Maximum Entropy Approach for the Prediction of Urban Mobility Patterns Simone DaniottiBernardo Monechi and Enrico Ubaldi Dated October 5 2022

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Maximum Entropy Approach for the Prediction of Urban Mobility Patterns

Simone Daniotti,∗Bernardo Monechi, and Enrico Ubaldi

(Dated: October 5, 2022)

The science of cities is a relatively new and interdisciplinary topic. It borrows techniques from

agent-based modeling, stochastic processes, and partial diﬀerential equations. However, how the

cities rise and fall, how they evolve, and the mechanisms responsible for these phenomena are still

open questions. Scientists have only recently started to develop forecasting tools, despite their

importance in urban planning, transportation planning, and epidemic spreading modeling. Here,

we build a fully interpretable statistical model that, incorporating only the minimum number of

constraints, can predict diﬀerent phenomena arising in the city. Using data on the movements

of car-sharing vehicles in diﬀerent Italian cities, we infer a model using the Maximum Entropy

(MaxEnt) principle. With it, we describe the activity in diﬀerent city zones and apply it to activity

forecasting and anomaly detection (e.g., strikes, and bad weather conditions). We compare our

method with diﬀerent models explicitly made for forecasting: SARIMA models and Deep Learning

Models. We ﬁnd that MaxEnt models are highly predictive, outperforming SARIMAs and having

similar results as a Neural Network. These results show how relevant statistical inference can be in

building a robust and general model describing urban systems phenomena. This article identiﬁes

the signiﬁcant observables for processes happening in the city, with the perspective of a deeper

understanding of the fundamental forces driving its dynamics.

INTRODUCTION

In recent years, pressing societal and environmental problems such as population growth, migrations, and climate

change, boosted the research on the Science of Cities and the related study of mobility. The availability of large

and detailed datasets covering the mobility of individuals at diﬀerent granularity contributed largely to enhance the

interest of researchers in this ﬁeld[1, 2]. Being multi-disciplinary, Science of Cities studies embrace diverse areas

of research. For example, statistical methods have been applied to city growth [3], multi-layer networks to urban

resilience [4] and spatial networks to describe and characterize the structure and the evolution of phenomena arising

on it [5]. On the modeling side, co-evolution models [6] and agent-based simulations [7] have been used to model

stylized facts and take into account policy making. Moreover, other frameworks such as ranking dynamics [8] have

been used to study urban environments and universal laws arising in them.

The mobility patterns of individuals diﬀusing in an urban environment are determined by the interplay of diﬀerent

mechanisms, such as the daily routines of the individuals and environmental constraints, both again regulated by even

more fundamental and interrelated phenomena, like wealth, trends, economic relations, socio-political disparities, and

cultural movements [9–11]. Urban environments are complex systems [12], and describing their growth and relation

with the surrounding cities is not unanimously understood by the scientiﬁc community [13]. Being that the commuting

phenomenon inside and between cities may be driven by diﬀerent events and may be related to diﬀerent causes, diﬀerent

tools and understandings must be developed [14, 15].

In this work, we study urban mobility patterns, building a model based on only a few constraints driven by

data observations. The model can predict diﬀerent events in the urban environment with high precision and can

be generalized to other dynamical processes unfolding in urban spaces. Here, we propose a Statistical Inference

(Maximum Entropy) approach to study and predict urban mobility patterns. Phenomena and relative observables

that happen inside the city are complex, they entangle with various indicators and are diﬃcult to predict. To build a

powerful, general, and robust statistical model, in principle we need to understand what are the important variables

that play a central role in the phenomenon we want to model. To solve it, we need to analyze the dataset we want to

study, identify the important dynamical properties and then model it solving the problem of optimizing the resulting

entropy.

The data represents the 30 −minute −binned multi-variate time-series of the activity of diﬀerent zones inside the

city.

Being that urban systems are notoriously complex and that the fundamental causes of the observed mobility patterns

are various and interrelated, our methodology is novel in this ﬁeld in the sense that builds the most general model

constrained to reproduce the correlations observed.

∗Complexity Science Hub Vienna, Vienna, 1080, Austria; daniotti@csh.ac.at

arXiv:2210.01491v1 [physics.soc-ph] 4 Oct 2022

In section Results we present the results for the Metropolitan City of Milan: multi-variate forecasting of the zones

activity and forecasting outliers events, discussed in Discussion. We also settle the basis for the formal derivation of

the model. We give an introduction to Maximum Entropy models, the formalism used in the following sessions and

the optimization algorithm used. Afterward,we introduce the formalism to compute an approximated Log-Likelihood.

In Additional Informations can be found similar results for the cities of Rome, Florence, and Turin.

In section Methods we describe the data and the variables in use. To ﬁnd out which are the important variables to

represent, we do a historical analysis and observe a dynamics of contraction and dilation of the time-shifted correlations

between diﬀerent zones. With this, we understand the important eﬀects and and ﬁnally we deﬁne and derive the

formulae for the ME model describing our data.

We use MaxEnt inference to obtain the parameters (using gradient ascent algorithm) and reproduce lag-correlations

of deﬁnite positive time series. We derive a model that is highly predictive at least as more sophisticated models that

take into account nonlinear correlations and have more parameters. We also use the obtained statistical model to ﬁnd

extrema events (outliers, such as strikes and bad weather days). As a result, we ﬁnd the dynamics of cars’ presence

in urban areas to be extremely rich and complex. We infer the couplings parameters between the activity proﬁles of

diﬀerent areas and use them to project the cars’ locations in time eﬃciently.

RESULTS

In this section, given a brief introduction to the method and the data studied, we present the results. For a more

in depth description, we refer to sec. Methods.

Data

The data we use is a normalised multivariate time-series representing the parking activity of the diﬀerent munici-

palities of the city. The procedure to obtain the ﬁnal form of the time-series in described in sec. Methods. In Fig. 1

we show an example of activity for two zones of our dataset for the Metropolitan City of Milan.

FIG. 1: Time series activity data for the city center (a) and for one of the suburbs (b).

Maximum Entropy Principle

The principle of maximum entropy states that the probability distribution which best represents the current state

of knowledge is the one with the largest entropy, in the context of precisely stated prior data. According to this

principle, the distribution with maximal information entropy is the best choice. The principle was ﬁrst shown by E.

T. Jaynes in two papers in the late ﬁfties where he emphasized a natural correspondence between statistical mechanics

and information theory [16, 17].

In particular, Jaynes oﬀered a new and very general rationale on why the Gibbsian method of statistical mechanics

works. He showed that statistical mechanics, and in particular Ensemble Theory, can be seen just as a particular

application of information theory, hence there is a strict correspondence between the entropy of statistical mechanics

and the Shannon information entropy.

Maximum Entropy models unveiled interesting results throughout the years for a large variety of systems, like

ﬂocking birds [18], proteins [19], brain [20] and social systems [21].

We will then implement this approach to deﬁne the model of our real-world system in the following sections. A more

general introduction to the maximum entropy formalism is out of scope here and we refer to the existing literature

for details [22–25].

The probability distribution with Maximum Entropy PME results from the extreme condition of the so-called

Generalized Entropy:

SP=SP+

k=1

θk(hOkiP(X)− hOkiobs),(1)

where

SP=−X

P(X) log(P(X)) (2)

is the Shannon Entropy of the probability distribution P(X). The maximum of the Generalized Entropy is the

maximum of the Entropy of the model when it is subject to constraints. Computing the functional-derivative (1) with

respect to P(X) and equating to zero results in:

Pme(X) = 1

Z(θ)exp h−

k=1

θkOk(X)i,(3)

where

Z(θ) = Z

Ω

dX exp h−

k=1

θkOk(X)i(4)

is the normalization (making a parallel with statistical physics, can be called Partition Function). Z(θ) is written

as a sum if Ω is discrete. Hence, the maximum entropy probability distribution is a Boltzmann distribution in the

canonical ensemble at temperature T= 1K, with eﬀective Hamiltonian H(X) = −PK

k=1 θkOk(X).

It must be noticed that the minimization of the generalized entropy is equivalent to the maximization of the

experimental average of the likelihood:

SP= log Z(θ)−

k=1

θkhOkie=−hlog Pmeie=1

m=1

log P(X(m)).(5)

In other words, the θkare chosen by imposing the experimental constraints on Entropy or, equivalently, by max-

imizing the global, experimental likelihood according to a model with the constraints cited above. Focusing on this

last sentence, we can say that the optimal parameters of θ(called eﬀective couplings) can be obtained through Max-

imum Likelihood, but only once one has assumed (by the principle of Maximum Entropy) that the most probable

distribution has the form of PM E .

Given the generative model probability distribution of conﬁgurations P(X|θ) and its corresponding partition

function by log Z(θ), the estimator of θcan be found by maximizing the log-likelihood:

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MaximumEntropyApproachforthePredictionofUrbanMobilityPatternsSimoneDaniotti,BernardoMonechi,andEnricoUbaldi(Dated:October5,2022)Thescienceofcitiesisarelativelynewandinterdisciplinarytopic.Itborrowstechniquesfromagent-basedmodeling,stochasticprocesses,andpartialdierentialequations.However,howthecit...

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Maximum Entropy Approach for the Prediction of Urban Mobility Patterns Simone DaniottiBernardo Monechi and Enrico Ubaldi Dated October 5 2022

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